Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ - x^{2} z + x y^{2} - x z^{2} + 2 z^{2} w - 2 z w^{2} $ |
| $=$ | $x^{2} z + x y^{2} + x z^{2} - 3 x z w - 2 z^{2} w - z w^{2}$ |
| $=$ | $2 x^{3} + x^{2} z - 3 x^{2} w - x y^{2} - x z^{2} - x z w + x w^{2} + 2 z^{2} w + z w^{2}$ |
| $=$ | $2 x^{3} + 2 x^{2} z - x^{2} w - 2 x y^{2} + x z w - 2 x w^{2} - z w^{2} + w^{3}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{3} z + 3 x^{2} y^{2} - x^{2} z^{2} - 6 x y^{2} z - 2 x z^{3} + z^{4} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -2x^{5} + 5x^{4} - 5x^{2} + 2x $ |
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This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:1:0:0)$, $(-1:0:1:1)$, $(-1:0:1:0)$, $(1:0:0:1)$, $(1/2:0:0:1)$, $(0:0:1:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{2^6}\cdot\frac{266240xz^{9}-1671168xz^{8}w-602112xz^{7}w^{2}-11218944xz^{6}w^{3}-16134144xz^{5}w^{4}-33293760xz^{4}w^{5}+22943040xz^{3}w^{6}+30053004xz^{2}w^{7}+6674433xzw^{8}+331760xw^{9}+65536y^{10}-4096y^{8}w^{2}+33792y^{6}w^{4}-25776y^{4}w^{6}+12987y^{2}w^{8}-65536z^{10}-1040384z^{9}w-178176z^{8}w^{2}-14733312z^{7}w^{3}-14360064z^{6}w^{4}-40175808z^{5}w^{5}+29978208z^{4}w^{6}+39152460z^{3}w^{7}+1326867z^{2}w^{8}-2400274zw^{9}-165872w^{10}}{w^{2}z^{2}(z-w)(64xz^{4}+176xz^{3}w+4xzw^{3}-xw^{4}+64z^{5}+208z^{4}w-8z^{3}w^{2}-20z^{2}w^{3}-2zw^{4}+w^{5})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
12.48.2.a.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{2}{3}y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 3X^{2}Y^{2}+2X^{3}Z-6XY^{2}Z-X^{2}Z^{2}-2XZ^{3}+Z^{4} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
12.48.2.a.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle \frac{2}{3}x-\frac{1}{3}w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{2}{9}x^{2}y+\frac{4}{9}xyw$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}x-\frac{2}{3}w$ |
This modular curve minimally covers the modular curves listed below.
This modular curve is minimally covered by the modular curves in the database listed below.
